# Basic Graph Theory

• Dieter Jungnickel
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 5)

## Abstract

Graph theory began in 1736 when Leonhard Euler solved the well-known Königsberg bridge problem. This problem asked for a circular walk through the town of Königsberg (now Kaliningrad) in such a way as to cross over each of the seven bridges spanning the river Pregel once, and only once. Euler realized that the precise shapes of the island and the other three territories involved are not important; the solvability depends only on their connection properties. This led to the abstract notion of a graph. Actually, Euler proved much more: he gave a necessary and sufficient condition for an arbitrary graph to admit such a circular walk. His theorem is one of the highlights in the introductory Chap. 1, where we deal with some of the most fundamental notions in graph theory: paths, cycles, connectedness, 1-factors, trees, Euler tours and Hamiltonian cycles, the travelling salesman problem, drawing graphs in the plane, and directed graphs. We will also see a first application, namely setting up a schedule for a tournament, say in soccer or basketball, where each of the 2n participating teams should play against each of the other teams exactly twice, once at home and once away.

## Keywords

Planar Graph Hamiltonian Cycle Euler Tour Hamiltonian Graph Start Vertex
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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